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Measures of Sections of Convex Bodies

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Convexity and Concentration

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 161))

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Abstract

This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.

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References

  1. M. Alexander, M. Henk and A. Zvavitch, A discrete version of Koldobsky’s slicing inequality, arXiv:1511.02702.

    Google Scholar 

  2. K. Ball, Some remarks on the geometry of convex sets, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math. 1317, Springer-Verlag, Berlin-Heidelberg-New York, 1988, 224–231.

    Google Scholar 

  3. K. Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), 891–901.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Ball, Logarithmically concave functions and sections of convex sets in \(\mathbb{R}^{n}\), Studia Math. 88 (1988), 69–84.

    MathSciNet  MATH  Google Scholar 

  5. K. Ball, Normed spaces with a weak-Gordon-Lewis property, Functional analysis (Austin, TX, 1987/1989), 36–47, Lecture Notes in Math., 1470, Springer, Berlin, 1991.

    Google Scholar 

  6. K. Ball, Isometric problems in ℓ p and sections of convex sets, Ph.D. dissertation, Trinity College, Cambridge (1986).

    Google Scholar 

  7. K. Ball and V. H. Nguyen, Entropy jumps for isotropic log-concave random vectors and spectral gap, Studia Math. 213 (2012), 81–96.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Bobkov and F. Nazarov, On convex bodies and log-concave probability measures with unconditional basis, Geometric aspects of functional analysis (Milman-Schechtman, eds), Lecture Notes in Math. 1807 (2003), 53–69.

    Article  Google Scholar 

  9. J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), 1467–1476.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bourgain, Geometry of Banach spaces and harmonic analysis, Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986), Amer. Math. Soc., Providence, RI, 1987, 871–878.

    Google Scholar 

  11. J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, Lecture Notes in Math. 1469 (1991), 127–137.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Bourgain, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. 1 (1991), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Bourgain, B. Klartag, V. Milman, Symmetrization and isotropic constants of convex bodies, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1850 (2004), 101–116.

    Article  MATH  Google Scholar 

  14. J. Bourgain, Gaoyong Zhang, On a generalization of the Busemann-Petty problem, Convex geometric analysis (Berkeley, CA, 1996), 65–76, Math. Sci. Res. Inst. Publ., 34, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  15. S. Brazitikos, A. Giannopoulos, P. Valettas and B. Vritsiou, Geometry of isotropic log-concave measures, Amer. Math. Soc., Providence, RI, 2014.

    MATH  Google Scholar 

  16. H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88–94.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Chasapis, A. Giannopoulos and D. Liakopoulos, Estimates for measures of lower dimensional sections of convex bodies, preprint; arXiv:1512.08393.

    Google Scholar 

  18. N. Dafnis and G. Paouris, Small ball probability estimates, ψ 2-behavior and the hyperplane conjecture, J. Funct. Anal. 258 (2010), 1933–1964.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Dar, Remarks on Bourgain’s problem on slicing of convex bodies, Operator theory, Advances and Applications 77 (1995), 61–66.

    MathSciNet  MATH  Google Scholar 

  20. R. Eldan and B. Klartag, Approximately gaussian marginals and the hyperplane conjecture, Proceedings of the Workshop on “Concentration, Functional Inequalities and Isoperimetry”, Contemp. Math. 545 (2011), 55–68.

    Article  MATH  Google Scholar 

  21. R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), 435–445.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Annals of Math. 140 (1994), 435–447.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. J. Gardner, Geometric tomography, Second edition, Cambridge University Press, Cambridge, 2006, 492 p.

    Book  MATH  Google Scholar 

  24. R.J. Gardner, A. Koldobsky, Th. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Annals of Math. 149 (1999), 691–703.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Giannopoulos, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239–244.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Giannopoulos, A. Koldobsky and P. Valettas, Inequalities for the surface area of projections of convex bodies, prerpint; arXiv:1601.05600.

    Google Scholar 

  27. A. Giannopoulos, G. Paouris and B. Vritsiou, A remark on the slicing problem, J. Funct. Anal. 262 (2012), 1062–1086.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Goodey and W. Weil, Intersection bodies and ellipsoids, Mathematika 42 (1958), 295–304.

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Goodey and Gaoyong Zhang, Inequalities between projection functions of convex bodies, Amer. J. Math. 120 (1998) 345–367.

    Google Scholar 

  30. E. Grinberg and Gaoyong Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. 78 (1999), 77–115.

    Google Scholar 

  31. M. Junge, On the hyperplane conjecture for quotient spaces of L p , Forum Math. 6 (1994), 617–635.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Junge, Proportional subspaces of spaces with unconditional basis have good volume properties, Geometric aspects of functional analysis (Israel Seminar, 1992–1994), 121–129, Oper. Theory Adv. Appl., 77, Birkhauser, Basel, 1995.

    Google Scholar 

  33. B. Klartag, An isomorphic version of the slicing problem, J. Funct. Anal. 218 (2005), 372–394.

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), 1274–1290.

    Article  MathSciNet  MATH  Google Scholar 

  35. B. Klartag and G. Kozma, On the hyperplane conjecture for random convex sets, Israel J. Math. 170 (2009), 253–268.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Koldobsky, Intersection bodies, positive definite distributions and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827–840.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Koldobsky, Intersection bodies in \(\mathbb{R}^{4}\), Adv. Math. 136 (1998), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000), 1507–1526.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Koldobsky, Fourier analysis in convex geometry, Amer. Math. Soc., Providence RI, 2005, 170 p.

    Book  MATH  Google Scholar 

  40. A. Koldobsky Stability in the Busemann-Petty and Shephard problems, Adv. Math. 228 (2011), 2145–2161.

    Google Scholar 

  41. A. Koldobsky, A hyperplane inequality for measures of convex bodies in \(\mathbb{R}^{n},\ n \leq 4\), Discrete Comput. Geom. 47 (2012), 538–547.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Koldobsky, Stability and separation in volume comparison problems, Math. Model. Nat. Phenom. 8 (2012), 159–169.

    MathSciNet  Google Scholar 

  43. A. Koldobsky, A \(\sqrt{n}\) estimate for measures of hyperplane sections of convex bodies, Adv. Math. 254 (2014), 33–40.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Koldobsky, Estimates for measures of sections of convex bodies, GAFA Seminar Volume, B.Klartag and E.Milman, editors, Lect. Notes in Math. 2116 (2014), 261–271.

    Google Scholar 

  45. A. Koldobsky, Slicing inequalities for subspaces of L p , Proc. Amer. Math. Soc. 144 (2016), 787–795.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Koldobsky, Slicing inequalities for measures of convex bodies, Adv. Math. 283 (2015), 473–488.

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Koldobsky, Isomorphic Busemann-Petty problem for sections of proportional dimensions, Adv. in Appl. Math. 71 (2015), 138–145.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Koldobsky, Stability inequalities for projections of convex bodies, Discrete Comput. Geom. 57 (2017), 152–163.

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Koldobsky and Dan Ma, Stability and slicing inequalities for intersection bodies, Geom. Dedicata 162 (2013), 325–335.

    Google Scholar 

  50. A. Koldobsky and A. Pajor, A remark on measures of sections of L p -balls, in: Geometric Aspects of Functional Analysis, Israel Seminar (2014–2016), ed. by B.Klartag and E.Milman. Lecture Notes in Math, to appear.

    Google Scholar 

  51. A. Koldobsky, A. Pajor and V. Yaskin, Inequalities of the Kahane-Khinchin type and sections of L p -balls, Studia Math. 184 (2008), 217–231.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Koldobsky, G. Paouris and M. Zymonopoulou, Isomorphic properties of intersection bodies, J. Funct. Anal. 261 (2011), 2697–2716.

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Koldobsky and A. Zvavitch, An isomorphic version of the Busemann-Petty problem for arbitrary measures, Geom. Dedicata 174 (2015), 261–277.

    Article  MathSciNet  MATH  Google Scholar 

  54. H. König, M. Meyer, A. Pajor, The isotropy constants of Schatten classes are bounded, Math. Ann. 312 (1998), 773–783.

    Article  MathSciNet  MATH  Google Scholar 

  55. D. G. Larman and C. A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164–175.

    Article  MathSciNet  MATH  Google Scholar 

  56. E. Lutwak, Intersection bodies and dual mixed volumes, Advances in Math. 71 (1988), 232–261.

    Article  MathSciNet  MATH  Google Scholar 

  57. E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), 566–598.

    Article  MathSciNet  MATH  Google Scholar 

  58. E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2006), 530–567.

    Article  MathSciNet  MATH  Google Scholar 

  59. E. Milman, On the mean-width of isotropic convex bodies and their associated L p -centroid bodies, preprint, arXiv:1402.0209.

    Google Scholar 

  60. V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Geometric Aspects of Functional Analysis, ed. by J. Lindenstrauss and V. D. Milman, Lecture Notes in Mathematics 1376, Springer, Heidelberg, 1989, pp. 64–104.

    Google Scholar 

  61. G. Paouris, On the isotropic constant of non-symmetric convex bodies, Geom. Aspects of Funct. Analysis. Israel Seminar 1996–2000, Lect. Notes in Math., 1745 (2000), 239–244.

    Google Scholar 

  62. M. Papadimitrakis, On the Busemann-Petty problem about convex, centrally symmetric bodies in \(\mathbb{R}^{n}\), Mathematika 39 (1992), 258–266

    Article  MathSciNet  MATH  Google Scholar 

  63. C. M. Petty, Projection bodies, Proc. Coll. Convexity (Copenhagen 1965), Kobenhavns Univ. Mat. Inst., 234–241.

    Google Scholar 

  64. B. Rubin and Gaoyong Zhang, Generalizations of the Busemann-Petty problem for sections of convex bodies, J. Funct. Anal. 213 (2004), 473–501.

    Google Scholar 

  65. R. Schneider, Zu einem problem von Shephard über die projektionen konvexer Körper, Math. Z. 101 (1967), 71–82.

    Article  MathSciNet  MATH  Google Scholar 

  66. G. C. Shephard, Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229–306.

    Article  MathSciNet  MATH  Google Scholar 

  67. Gaoyong Zhang, Intersection bodies and Busemann-Petty inequalities in \(\mathbb{R}^{4}\), Annals of Math. 140 (1994), 331–346.

    Article  MathSciNet  MATH  Google Scholar 

  68. Gaoyong Zhang, A positive answer to the Busemann-Petty problem in four dimensions, Annals of Math. 149 (1999), 535–543.

    Article  MathSciNet  Google Scholar 

  69. Gaoyong Zhang, Sections of convex bodies, Amer. J. Math. 118 (1996), 319–340.

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Zvavitch, The Busemann-Petty problem for arbitrary measures, Math. Ann. 331 (2005), 867–887.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the US National Science Foundation grant DMS-1265155.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Alexander Koldobsky

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  1. Alexander Koldobsky
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Correspondence to Alexander Koldobsky .

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Editors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, New Jersey, USA

    Eric Carlen

  2. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    Mokshay Madiman

  3. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA

    Elisabeth M. Werner

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Koldobsky, A. (2017). Measures of Sections of Convex Bodies. In: Carlen, E., Madiman, M., Werner, E. (eds) Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7005-6_18

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